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I am reading a book on quantum field theory, while I have never been trained as a physicist. I found a big gap in language and have trouble understanding what physicists mean by "quantum field".

If I understand correctly, after quantizing twice (field and operator). quantum field should be an operator valued distribution. Am I right? I would appreciate it if some mathematician who are familiar with physics could kindly explain "quantum field" in terms of mathematics.

Thank you in advance.

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I am not sure what you math level is, so I'll try to make it as simple as possible. In standard quantum mechanics, we formalize the state of a physical system as belonging to Hilbert space. Hilbert space is basically a multidimensional space in which the number of dimensions are not discrete (such as in 3d space), but continuous. You can intuitively think of any real valued function f as a vector in Hilbert space, where the value f(x) is the coordinate of vector f along dimension x. The Schrödinger equation describes the evolution of the "quantum state" (a vector in Hilbert space). In second quantification you go a further step of abstraction. Now the "quantum field" is a mathematical object that belongs to Fock space. If it means something to you, Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H. The state of the quantum field allows you to calculate the distribution of values for repeated measurements on the system (using additional rules).

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